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{{Short description|Repeated oscillation around equilibrium}}
{{About|waves in the scientific sense|waves on seas and lakes|Wind wave|the human hand gesture|Waving|other uses|Wave (disambiguation)|and|Wave motion (disambiguation)}}
[[File:2006-01-14 Surface waves.jpg|thumb|Surface waves in water showing water ripples]]

In [[physics]], [[mathematics]], [[engineering]], and related fields, a '''wave''' is a propagating dynamic disturbance (change from [[List of types of equilibrium|equilibrium]]) of one or more [[quantities]]. ''[[Periodic wave]]s'' oscillate repeatedly about an equilibrium (resting) value at some [[frequency]]. When the entire [[waveform]] moves in one direction, it is said to be a '''travelling wave'''; by contrast, a pair of [[superposition principle|superimposed]] periodic waves traveling in opposite directions makes a ''[[standing wave]]''. In a standing wave, the amplitude of vibration has nulls at some positions where the wave amplitude appears smaller or even zero.

There are two types of waves that are most commonly studied in [[classical physics]]: [[mechanical wave]]s and [[electromagnetic wave]]s. In a mechanical wave, [[Stress (mechanics)|stress]] and [[Strain (mechanics)|strain]] fields oscillate about a mechanical equilibrium. A mechanical wave is a local [[deformation (physics)|deformation (strain)]] in some physical medium that propagates from particle to particle by creating local [[stress (mechanics)|stresses]] that cause strain in neighboring particles too. For example, [[sound]] waves are variations of the local [[Sound pressure|pressure]] and [[Particle velocity|particle motion]] that propagate through the medium. Other examples of mechanical waves are [[seismic wave]]s, [[gravity wave]]s, [[surface wave]]s and [[string vibration]]s. In an electromagnetic wave (such as light), coupling between the electric and magnetic fields sustains propagation of waves involving these fields according to [[Maxwell's equations]]. Electromagnetic waves can travel through a [[vacuum]] and through some [[dielectric]] media (at wavelengths where they are considered [[transparency and translucency|transparent]]). Electromagnetic waves, as determined by their frequencies (or [[wavelength]]s), have more specific designations including [[radio wave]]s, [[infrared radiation]], [[terahertz waves]], [[visible light]], [[ultraviolet radiation]], [[X-ray]]s and [[gamma ray]]s.

Other types of waves include [[gravitational wave]]s, which are disturbances in [[spacetime]] that propagate according to [[general relativity]]; [[heat equation|heat diffusion waves]]; [[plasma wave]]s that combine mechanical deformations and electromagnetic fields; [[reaction–diffusion system|reaction–diffusion waves]], such as in the [[Belousov–Zhabotinsky reaction]]; and many more. Mechanical and electromagnetic waves transfer [[energy]],<ref>{{Harv|Hall|1980| p=8}}</ref> [[momentum (physics)|momentum]], and [[information]], but they do not transfer particles in the medium. In mathematics and [[electronics]] waves are studied as [[signal]]s.<ref>Pragnan Chakravorty, "What Is a Signal? [Lecture Notes]",&nbsp;IEEE ''Signal Processing Magazine'', vol. 35, no. 5, pp. 175–177, Sept. 2018.&nbsp;{{doi|10.1109/MSP.2018.2832195}}</ref> On the other hand, some waves have [[Envelope (waves)|envelopes]] which do not move at all such as [[standing wave]]s (which are fundamental to music) and [[hydraulic jump]]s.

[[File:Santos E et al Neuroimage 2014 .gif|thumb|Example of biological waves expanding over the brain cortex, an example of [[Cortical spreading depression|spreading depolarizations]].<ref>{{Cite journal|last1=Santos|first1=Edgar|last2=Schöll|first2=Michael|last3=Sánchez-Porras|first3=Renán|last4=Dahlem|first4=Markus A.|last5=Silos|first5=Humberto|last6=Unterberg|first6=Andreas|last7=Dickhaus|first7=Hartmut|last8=Sakowitz|first8=Oliver W.|date=2014-10-01|title=Radial, spiral and reverberating waves of spreading depolarization occur in the gyrencephalic brain|journal=NeuroImage|volume=99|pages=244–255|doi=10.1016/j.neuroimage.2014.05.021|issn=1095-9572|pmid=24852458|s2cid=1347927}}</ref>]]

A physical wave [[field (physics)|field]] is almost always confined to some finite region of space, called its ''domain''. For example, the seismic waves generated by [[earthquakes]] are significant only in the interior and surface of the planet, so they can be ignored outside it. However, waves with infinite domain, that extend over the whole space, are commonly studied in mathematics, and are very valuable tools for understanding physical waves in finite domains.

A ''[[plane wave]]'' is an important mathematical idealization where the disturbance is identical along any (infinite) plane [[Normal (geometry)|normal]] to a specific direction of travel. Mathematically, the simplest wave is a ''[[sinusoidal plane wave]]'' in which at any point the field experiences [[simple harmonic motion]] at one frequency. In linear media, complicated waves can generally be decomposed as the sum of many sinusoidal plane waves having [[Angular spectrum method|different directions of propagation]] and/or [[Fourier transform|different frequencies]]. A plane wave is classified as a ''[[transverse wave]]'' if the field disturbance at each point is described by a vector perpendicular to the direction of propagation (also the direction of energy transfer); or ''[[longitudinal wave]]'' if those vectors are aligned with the propagation direction. Mechanical waves include both transverse and longitudinal waves; on the other hand electromagnetic plane waves are strictly transverse while sound waves in fluids (such as air) can only be longitudinal. That physical direction of an oscillating field relative to the propagation direction is also referred to as the wave's ''[[polarization (waves)|polarization]]'', which can be an important attribute.
{{Modern physics}}

== Mathematical description ==

=== Single waves ===
{{See also|Solitary wave (disambiguation){{!}}Solitary wave}}
A wave can be described just like a field, namely as a [[function (mathematics)|function]] <math>F(x,t)</math> where <math>x</math> is a position and <math>t</math> is a time.

The value of <math>x</math> is a point of space, specifically in the region where the wave is defined. In mathematical terms, it is usually a [[vector (mathematics)|vector]] in the [[analytic geometry|Cartesian three-dimensional space]] <math>\mathbb{R}^3</math>. However, in many cases one can ignore one dimension, and let <math>x</math> be a point of the Cartesian plane <math>\mathbb{R}^2</math>. This is the case, for example, when studying vibrations of a drum skin. One may even restrict <math>x</math> to a point of the Cartesian line <math>\R</math> – that is, the set of [[real number]]s. This is the case, for example, when studying vibrations in a [[string (music)|violin string]] or [[recorder (musical instrument)|recorder]]. The time <math>t</math>, on the other hand, is always assumed to be a [[scalar (physics)|scalar]]; that is, a real number.

The value of <math>F(x,t)</math> can be any physical quantity of interest assigned to the point <math>x</math> that may vary with time. For example, if <math>F</math> represents the vibrations inside an elastic solid, the value of <math>F(x,t)</math> is usually a vector that gives the current displacement from <math>x</math> of the material particles that would be at the point <math>x</math> in the absence of vibration. For an electromagnetic wave, the value of <math>F</math> can be the [[electric field]] vector <math>E</math>, or the [[magnetic field]] vector <math>H</math>, or any related quantity, such as the [[Poynting vector]] <math>E\times H</math>. In [[fluid dynamics]], the value of <math>F(x,t)</math> could be the velocity vector of the fluid at the point <math>x</math>, or any scalar property like [[pressure]], [[temperature]], or [[density]]. In a chemical reaction, <math>F(x,t)</math> could be the concentration of some substance in the neighborhood of point <math>x</math> of the reaction medium.

For any dimension <math>d</math> (1, 2, or 3), the wave's domain is then a [[subset]] <math>D</math> of <math>\mathbb{R}^d</math>, such that the function value <math>F(x,t)</math> is defined for any point <math>x</math> in <math>D</math>. For example, when describing the motion of a [[drumhead|drum skin]], one can consider <math>D</math> to be a [[disk (mathematics)|disk]] (circle) on the plane <math>\mathbb{R}^2</math> with center at the origin <math>(0,0)</math>, and let <math>F(x,t)</math> be the vertical displacement of the skin at the point <math>x</math> of <math>D</math> and at time <math>t</math>.

=== Superposition ===
{{main|Superposition principle}}
Waves of the same type are often superposed and encountered simultaneously at a given point in space and time. The properties at that point are the sum of the properties of each component wave at that point. In general, the velocities are not the same, so the wave form will change over time and space.

=== Wave spectrum ===
{{See also|Wind wave#Spectrum|Electromagnetic spectrum|Spectrum (physical sciences)}}
{{Expand section|concept summary|date=May 2023}}

=== Wave families ===
Sometimes one is interested in a single specific wave. More often, however, one needs to understand large set of possible waves; like all the ways that a drum skin can vibrate after being struck once with a [[drum stick]], or all the possible [[radar]] echoes one could get from an [[airplane]] that may be approaching an [[airport]].

In some of those situations, one may describe such a family of waves by a function <math>F(A,B,\ldots;x,t)</math> that depends on certain [[parameter]]s <math>A,B,\ldots</math>, besides <math>x</math> and <math>t</math>. Then one can obtain different waves – that is, different functions of <math>x</math> and <math>t</math> – by choosing different values for those parameters.

[[File:Half-open pipe wave.gif|class=skin-invert-image|thumb|right|Sound pressure standing wave in a half-open pipe playing the 7th harmonic of the fundamental (''n'' = 4)]]
For example, the sound pressure inside a [[recorder (musical instrument)|recorder]] that is playing a "pure" note is typically a [[standing wave]], that can be written as
: <math>F(A,L,n,c;x,t) = A \left(\cos 2\pi x\frac{2 n - 1}{4 L}\right) \left(\cos 2\pi c t\frac{2n - 1}{4 L}\right)</math>
The parameter <math>A</math> defines the amplitude of the wave (that is, the maximum sound pressure in the bore, which is related to the loudness of the note); <math>c</math> is the speed of sound; <math>L</math> is the length of the bore; and <math>n</math> is a positive integer (1,2,3,...) that specifies the number of [[standing wave|nodes]] in the standing wave. (The position <math>x</math> should be measured from the [[wind instrument|mouthpiece]], and the time <math>t</math> from any moment at which the pressure at the mouthpiece is maximum. The quantity <math>\lambda = 4L/(2 n - 1)</math> is the [[wavelength]] of the emitted note, and <math>f = c/\lambda</math> is its [[frequency]].) Many general properties of these waves can be inferred from this general equation, without choosing specific values for the parameters.

As another example, it may be that the vibrations of a drum skin after a single strike depend only on the distance <math>r</math> from the center of the skin to the strike point, and on the strength <math>s</math> of the strike. Then the vibration for all possible strikes can be described by a function <math>F(r,s;x,t)</math>.

Sometimes the family of waves of interest has infinitely many parameters. For example, one may want to describe what happens to the temperature in a metal bar when it is initially heated at various temperatures at different points along its length, and then allowed to cool by itself in vacuum. In that case, instead of a scalar or vector, the parameter would have to be a function <math>h</math> such that <math>h(x)</math> is the initial temperature at each point <math>x</math> of the bar. Then the temperatures at later times can be expressed by a function <math>F</math> that depends on the function <math>h</math> (that is, a [[operator (mathematics)|functional operator]]), so that the temperature at a later time is <math>F(h;x,t)</math>

=== Differential wave equations ===
Another way to describe and study a family of waves is to give a mathematical equation that, instead of explicitly giving the value of <math>F(x,t)</math>, only constrains how those values can change with time. Then the family of waves in question consists of all functions <math>F</math> that satisfy those constraints – that is, all [[solution (mathematics)|solutions]] of the equation.

This approach is extremely important in physics, because the constraints usually are a consequence of the physical processes that cause the wave to evolve. For example, if <math>F(x,t)</math> is the temperature inside a block of some [[homogeneous]] and [[isotropic]] solid material, its evolution is constrained by the [[partial differential equation]]
: <math>\frac{\partial F}{\partial t}(x,t) = \alpha \left(\frac{\partial^2 F}{\partial x_1^2}(x,t) + \frac{\partial^2 F}{\partial x_2^2}(x,t) + \frac{\partial^2 F}{\partial x_3^2}(x,t) \right) + \beta Q(x,t)</math>
where <math>Q(p,f)</math> is the heat that is being generated per unit of volume and time in the neighborhood of <math>x</math> at time <math>t</math> (for example, by chemical reactions happening there); <math>x_1,x_2,x_3</math> are the Cartesian coordinates of the point <math>x</math>; <math>\partial F/\partial t</math> is the (first) derivative of <math>F</math> with respect to <math>t</math>; and <math>\partial^2 F/\partial x_i^2</math> is the second derivative of <math>F</math> relative to <math>x_i</math>. (The symbol "<math>\partial</math>" is meant to signify that, in the derivative with respect to some variable, all other variables must be considered fixed.)

This equation can be derived from the laws of physics that govern the [[heat diffusion|diffusion of heat]] in solid media. For that reason, it is called the [[heat equation]] in mathematics, even though it applies to many other physical quantities besides temperatures.

For another example, we can describe all possible sounds echoing within a container of gas by a function <math>F(x,t)</math> that gives the pressure at a point <math>x</math> and time <math>t</math> within that container. If the gas was initially at uniform temperature and composition, the evolution of <math>F</math> is constrained by the formula
: <math>\frac{\partial^2 F}{\partial t^2}(x,t) = \alpha \left(\frac{\partial^2 F}{\partial x_1^2}(x,t) + \frac{\partial^2 F}{\partial x_2^2}(x,t) + \frac{\partial^2 F}{\partial x_3^2}(x,t) \right) + \beta P(x,t)</math>
Here <math>P(x,t)</math> is some extra compression force that is being applied to the gas near <math>x</math> by some external process, such as a [[loudspeaker]] or [[piston]] right next to <math>p</math>.

This same differential equation describes the behavior of mechanical vibrations and electromagnetic fields in a homogeneous isotropic non-conducting solid. Note that this equation differs from that of heat flow only in that the left-hand side is <math>\partial^2 F/\partial t^2</math>, the second derivative of <math>F</math> with respect to time, rather than the first derivative <math>\partial F/\partial t</math>. Yet this small change makes a huge difference on the set of solutions <math>F</math>. This differential equation is called "the" [[wave equation]] in mathematics, even though it describes only one very special kind of waves.

== Wave in elastic medium ==
<!--Still needs substantial cleanup below this point-->
{{Main|Wave equation|d'Alembert's formula}}

Consider a traveling [[transverse wave]] (which may be a [[pulse (physics)|pulse]]) on a string (the medium). Consider the string to have a single spatial dimension. Consider this wave as traveling

[[File:Nonsinusoidal wavelength.svg|class=skin-invert-image|thumb|right|200 px|Wavelength ''λ'' can be measured between any two corresponding points on a waveform.]]
[[File:Superpositionprinciple.gif|class=skin-invert-image|thumb|Animation of two waves, the green wave moves to the right while blue wave moves to the left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves. Note that {{nowrap|1=''f''(''x'', ''t'') + ''g''(''x'', ''t'') = ''u''(''x'', ''t'')}}. |left]]
* in the <math>x</math> direction in space. For example, let the positive <math>x</math> direction be to the right, and the negative <math>x</math> direction be to the left.
* with constant [[amplitude]] <math>u</math>
* with constant velocity <math>v</math>, where <math>v</math> is
** independent of [[wavelength]] (no [[dispersion relation|dispersion]])
** independent of amplitude ([[linear]] media, not [[Nonlinearity|nonlinear]]).<ref name=Helbig>{{cite book |title = Seismic waves and rays in elastic media |chapter-url = https://books.google.com/books?id=s7bp6ezoRhcC&pg=PA134 |pages = 131 ''ff'' |author = Michael A. Slawinski |chapter = Wave equations |isbn = 978-0-08-043930-3 |year = 2003 |publisher = Elsevier }}</ref><ref name=Ostrovsky>{{cite book |title = Modulated waves: theory and application |url = https://www.amazon.com/gp/product/0801873258 |author1=Lev A. Ostrovsky |author2=Alexander I. Potapov |name-list-style=amp |publisher = Johns Hopkins University Press |isbn = 978-0-8018-7325-6 |year = 2001 }}</ref>
* with constant [[waveform]], or shape

This wave can then be described by the two-dimensional functions
{{unbulleted list | style = padding-left: 1.5em
| <math>u(x,t) = F(x - v t)</math> (waveform <math>F</math> traveling to the right)
| <math>u(x,t) = G(x + v t)</math> (waveform <math>G</math> traveling to the left)
}}
or, more generally, by [[d'Alembert's formula]]:<ref name=Graaf>{{cite book |title = Wave motion in elastic solids |last = Graaf |first= Karl F |edition = Reprint of Oxford 1975 |publisher = Dover |year = 1991 |url = https://books.google.com/books?id=5cZFRwLuhdQC |pages = 13–14 |isbn = 978-0-486-66745-4 }}</ref>
<math display="block">u(x,t) = F(x - vt) + G(x + vt). </math>
representing two component waveforms <math>F</math> and <math>G</math> traveling through the medium in opposite directions. A generalized representation of this wave can be obtained<ref>For an example derivation, see the steps leading up to eq. (17) in {{cite web |url = http://prism.texarkanacollege.edu/physicsjournal/wave-eq.html |title = Kinematic Derivation of the Wave Equation |last = Redfern |first= Francis |work = Physics Journal |access-date = 2012-12-11 |archive-date = 2013-07-24 |archive-url = https://web.archive.org/web/20130724011045/http://prism.texarkanacollege.edu/physicsjournal/wave-eq.html |url-status = dead }}</ref> as the [[partial differential equation]]
<math display="block">\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}.</math>

General solutions are based upon [[Duhamel's principle]].<ref name=Struwe>{{cite book |title = Geometric wave equations |author1=Jalal M. Ihsan Shatah |author2=Michael Struwe |chapter-url = https://books.google.com/books?id=zsasG2axbSoC&pg=PA37 |chapter = The linear wave equation |pages = 37''ff'' |isbn = 978-0-8218-2749-9 |year = 2000 |publisher = American Mathematical Society Bookstore }}</ref>

=== Wave forms ===
{{main|Waveform}}
[[File:Waveforms.svg|class=skin-invert-image|thumb|right|280 px|[[Sine wave|Sine]], [[Square wave|square]], [[Triangle wave|triangle]] and [[Sawtooth wave|sawtooth]] waveforms]]

The form or shape of ''F'' in [[d'Alembert's formula]] involves the argument ''x'' − ''vt''. Constant values of this argument correspond to constant values of ''F'', and these constant values occur if ''x'' increases at the same rate that ''vt'' increases. That is, the wave shaped like the function ''F'' will move in the positive ''x''-direction at velocity ''v'' (and ''G'' will propagate at the same speed in the negative ''x''-direction).<ref name=Lyons>{{cite book |url = https://books.google.com/books?id=WdPGzHG3DN0C&pg=PA128 |pages = 128 ''ff'' |title = All you wanted to know about mathematics but were afraid to ask |author = Louis Lyons |isbn = 978-0-521-43601-4 |publisher = Cambridge University Press |year = 1998 }}</ref>

In the case of a periodic function ''F'' with period ''λ'', that is, ''F''(''x'' + ''λ'' − ''vt'') = ''F''(''x'' − ''vt''), the periodicity of ''F'' in space means that a snapshot of the wave at a given time ''t'' finds the wave varying periodically in space with period ''λ'' (the [[wavelength]] of the wave). In a similar fashion, this periodicity of ''F'' implies a periodicity in time as well: ''F''(''x'' − ''v''(''t'' + ''T'')) = ''F''(''x'' − ''vt'') provided ''vT'' = ''λ'', so an observation of the wave at a fixed location ''x'' finds the wave undulating periodically in time with period ''T'' = ''λ''/''v''.<ref name="McPherson0">{{cite book |title = Introduction to Macromolecular Crystallography |last = McPherson |first= Alexander |chapter-url = https://books.google.com/books?id=o7sXm2GSr9IC&pg=PA77 |page = 77 |chapter = Waves and their properties |isbn = 978-0-470-18590-2 |year = 2009 |edition = 2 |publisher = Wiley }}</ref>

=== Amplitude and modulation ===
{{Main|Amplitude modulation}}
{{See also|Frequency modulation|Phase modulation}}

[[File:Amplitudemodulation.gif|class=skin-invert-image|thumb|Amplitude modulation can be achieved through ''f''(''x'',''t'') = 1.00×sin(2π/0.10×(''x''−1.00×''t'')) and ''g''(''x'',''t'') = 1.00×sin(2π/0.11×(''x''−1.00×''t'')) only the resultant is visible to improve clarity of waveform.|left]]
[[File:Wave packet.svg|class=skin-invert-image|right|thumb|Illustration of the ''envelope'' (the slowly varying red curve) of an amplitude-modulated wave. The fast varying blue curve is the ''carrier'' wave, which is being modulated.]]

The amplitude of a wave may be constant (in which case the wave is a ''c.w.'' or ''[[continuous wave]]''), or may be ''modulated'' so as to vary with time and/or position. The outline of the variation in amplitude is called the ''envelope'' of the wave. Mathematically, the [[Amplitude modulation|modulated wave]] can be written in the form:<ref name=Jirauschek>{{cite book |url = https://books.google.com/books?id=6kOoT_AX2CwC&pg=PA9 |page = 9 |title = FEW-cycle Laser Dynamics and Carrier-envelope Phase Detection |author = Christian Jirauschek |isbn = 978-3-86537-419-6 |year = 2005 |publisher = Cuvillier Verlag }}</ref><ref name="Kneubühl">{{cite book |title = Oscillations and waves |author = Fritz Kurt Kneubühl |url = https://books.google.com/books?id=geYKPFoLgoMC&pg=PA365 |page = 365 |year = 1997 |isbn = 978-3-540-62001-3 |publisher = Springer }}</ref><ref name=Lundstrom>{{cite book |url = https://books.google.com/books?id=FTdDMtpkSkIC&pg=PA33 |page = 33 |author = Mark Lundstrom |isbn = 978-0-521-63134-1 |year = 2000 |title = Fundamentals of carrier transport |publisher = Cambridge University Press }}</ref>
<math display="block">u(x,t) = A(x,t) \sin \left(kx - \omega t + \phi \right) , </math>
where <math>A(x,\ t)</math> is the amplitude envelope of the wave, <math>k</math> is the ''[[wavenumber]]'' and <math>\phi</math> is the ''[[phase (waves)|phase]]''. If the [[group velocity]] <math>v_g</math> (see below) is wavelength-independent, this equation can be simplified as:<ref name=Chen>{{cite book |chapter-url = https://books.google.com/books?id=LxzWPskhns0C&pg=PA363 |author = Chin-Lin Chen |title = Foundations for guided-wave optics |page = 363 |chapter = §13.7.3 Pulse envelope in nondispersive media |isbn = 978-0-471-75687-3 |year = 2006 |publisher = Wiley }}</ref>
<math display="block">u(x,t) = A(x - v_g t) \sin \left(kx - \omega t + \phi \right) , </math>
showing that the envelope moves with the group velocity and retains its shape. Otherwise, in cases where the group velocity varies with wavelength, the pulse shape changes in a manner often described using an ''envelope equation''.<ref name=Chen/><ref name="Recami">{{cite book |last1=Longhi |first1=Stefano |title=Localized Waves |last2=Janner |first2=Davide |publisher=Wiley-Interscience |year=2008 |isbn=978-0-470-10885-7 |editor1=Hugo E. Hernández-Figueroa |page=329 |chapter=Localization and Wannier wave packets in photonic crystals |editor2=Michel Zamboni-Rached |editor3=Erasmo Recami |chapter-url=https://books.google.com/books?id=xxbXgL967PwC&pg=PA329}}</ref>

=== Phase velocity and group velocity ===
{{Main|Phase velocity|Group velocity}}
{{See also|Envelope (waves)#Phase and group velocity}}
[[Image:Wave group.gif|class=skin-invert-image|thumb|The red square moves with the [[phase velocity]], while the green circles propagate with the [[group velocity]].|left]]

There are two velocities that are associated with waves, the [[phase velocity]] and the [[group velocity]].

Phase velocity is the rate at which the [[phase (waves)|phase]] of the wave [[Wave propagation|propagates in space]]: any given phase of the wave (for example, the [[crest (physics)|crest]]) will appear to travel at the phase velocity. The phase velocity is given in terms of the [[wavelength]] {{mvar|λ}} (lambda) and [[Wave period|period]] {{mvar|T}} as
<math display="block">v_\mathrm{p} = \frac{\lambda}{T}.</math>

[[Image:Wave opposite-group-phase-velocity.gif|class=skin-invert-image|thumb|right|A wave with the group and phase velocities going in different directions]]

Group velocity is a property of waves that have a defined envelope, measuring propagation through space (that is, phase velocity) of the overall shape of the waves' amplitudes—modulation or envelope of the wave.

== Special waves ==

=== Sine waves ===
{{excerpt|Sine wave}}

=== Plane waves ===
{{Main|Plane wave}}

A [[plane wave]] is a kind of wave whose value varies only in one spatial direction. That is, its value is constant on a plane that is perpendicular to that direction. Plane waves can be specified by a vector of unit length <math>\hat n</math> indicating the direction that the wave varies in, and a wave profile describing how the wave varies as a function of the displacement along that direction (<math>\hat n \cdot \vec{x}</math>) and time (<math>t</math>). Since the wave profile only depends on the position <math>\vec{x}</math> in the combination <math>\hat n \cdot \vec{x}</math>, any displacement in directions perpendicular to <math>\hat n</math> cannot affect the value of the field.

Plane waves are often used to model [[electromagnetic waves]] far from a source. For electromagnetic plane waves, the electric and magnetic fields themselves are transverse to the direction of propagation, and also perpendicular to each other.

=== Standing waves ===
{{Main|Standing wave|Acoustic resonance|Helmholtz resonance|Organ pipe}}

[[File:Standing wave.gif|class=skin-invert-image|thumb|right|300px|Standing wave. The red dots represent the wave [[Node (physics)|nodes]].]]

A standing wave, also known as a ''stationary wave'', is a wave whose [[Envelope (waves)|envelope]] remains in a constant position. This phenomenon arises as a result of [[Interference (wave propagation)|interference]] between two waves traveling in opposite directions.

The ''sum'' of two counter-propagating waves (of equal amplitude and frequency) creates a ''standing wave''. Standing waves commonly arise when a boundary blocks further propagation of the wave, thus causing wave reflection, and therefore introducing a counter-propagating wave. For example, when a [[violin]] string is displaced, transverse waves propagate out to where the string is held in place at the [[Bridge (instrument)|bridge]] and the [[Nut (string instrument)|nut]], where the waves are reflected back. At the bridge and nut, the two opposed waves are in [[antiphase]] and cancel each other, producing a [[node (physics)|node]]. Halfway between two nodes there is an [[antinode]], where the two counter-propagating waves ''enhance'' each other maximally. There is no net [[Energy transfer|propagation of energy]] over time.

<gallery>
Image:Standing waves on a string.gif|One-dimensional standing waves; the [[fundamental frequency|fundamental]] mode and the first 5 [[overtone]]s
Image:Drum vibration mode01.gif|A two-dimensional [[Vibrations of a circular drum|standing wave on a disk]]; this is the fundamental mode.
Image:Drum vibration mode21.gif|A [[Vibrations of a circular drum|standing wave on a disk]] with two nodal lines crossing at the center; this is an overtone.
</gallery>

=== Solitary waves ===
{{Main|Soliton}}
[[File:Soliton hydro.jpg|thumb|[[Solitary wave (water waves)|Solitary wave]] in a laboratory [[wave channel]]]]
A '''soliton''' or '''solitary wave''' is a self-reinforcing [[wave packet]] that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of [[nonlinearity|nonlinear]] and [[dispersion relation|dispersive effects]] in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersive [[partial differential equation]]s describing physical systems.

== Physical properties ==

=== Propagation===

Wave propagation is any of the ways in which waves travel. With respect to the direction of the [[oscillation]] relative to the propagation direction, we can distinguish between [[longitudinal wave]] and [[transverse wave]]s.

[[Electromagnetic wave]]s propagate in [[vacuum]] as well as in material media. Propagation of other wave types such as sound may occur only in a [[transmission medium]].

==== Reflection of plane waves in a half-space  ====
{{further|Reflection coefficient}}
The propagation and reflection of plane waves—e.g. Pressure waves ([[P wave]]) or [[S-wave|Shear waves (SH or SV-waves)]] are phenomena that were first characterized within the field of classical seismology, and are now considered fundamental concepts in modern [[seismic tomography]]. The analytical solution to this problem exists and is well known. The frequency domain solution can be obtained by first finding the [[Helmholtz decomposition]] of the displacement field, which is then substituted into the [[wave equation]]. From here, the [[Wave equation#Plane-wave eigenmodes|plane wave eigenmodes]] can be calculated.{{citation needed|date=May 2023}}{{clarify|Unintelligible to the ordinary reader|date=May 2023}}

==== SV wave propagation ====
[[File:SV wave propagation.gif|thumb|upright=1.4|The propagation of SV-wave in a homogeneous half-space (the horizontal displacement field)]]
[[File:SV wave propagation y.gif|thumb|upright=1.4|The propagation of SV-wave in a homogeneous half-space (The vertical displacement field){{clarify|date=May 2023}}]]
The analytical solution of SV-wave in a half-space indicates that the plane SV wave reflects back to the domain as a P and SV waves, leaving out special cases. The angle of the reflected SV wave is identical to the incidence wave, while the angle of the reflected P wave is greater than the SV wave. For the same wave frequency, the SV wavelength is smaller than the P wavelength. This fact has been depicted in this animated picture.<ref>The animations are taken from {{cite web
 | url = https://sites.google.com/a/utexas.edu/babakpoursartip/research
 | title = Topographic amplification of seismic waves
 | last = Poursartip
 | first = Babak
 | year = 2015
 | department = UT Austin
 | access-date = 2023-02-24
 | archive-date = 2017-01-09
 | archive-url = https://web.archive.org/web/20170109184014/https://sites.google.com/a/utexas.edu/babakpoursartip/research
 | url-status = dead
}}</ref>

==== P wave propagation ====
Similar to the SV wave, the P incidence, in general, reflects as the P and SV wave. There are some special cases where the regime is different.{{clarify|unintelligible to the ordinary reader|date=May 2023}}

=== Wave velocity ===
{{further|Phase velocity|Group velocity|Signal velocity}}
[[Image:Seismic wave prop mine.gif|thumb|upright=1.4|[[Seismic wave]] propagation in 2D modelled using [[FDTD]] method in the presence of a landmine]]

Wave velocity is a general concept, of various kinds of wave velocities, for a wave's [[phase (waves)|phase]] and [[speed]] concerning energy (and information) propagation. The [[phase velocity]] is given as:
<math display="block">v_{\rm p} = \frac{\omega}{k},</math>
where:
* ''v''<sub>p</sub> is the phase velocity (with SI unit m/s),
* ''ω'' is the [[angular frequency]] (with SI unit rad/s),
* ''k'' is the [[wavenumber]] (with SI unit rad/m).

The phase speed gives you the speed at which a point of constant [[phase (waves)|phase]] of the wave will travel for a discrete frequency. The angular frequency ''ω'' cannot be chosen independently from the wavenumber ''k'', but both are related through the [[dispersion relation]]ship:
<math display="block">\omega = \Omega(k).</math>

In the special case {{math|1=Ω(''k'') = ''ck''}}, with ''c'' a constant, the waves are called non-dispersive, since all frequencies travel at the same phase speed ''c''. For instance [[electromagnetic wave]]s in [[vacuum]] are non-dispersive. In case of other forms of the dispersion relation, we have dispersive waves. The dispersion relationship depends on the medium through which the waves propagate and on the type of waves (for instance [[electromagnetic wave|electromagnetic]], [[sound wave|sound]] or [[ocean surface wave|water]] waves).

The speed at which a resultant [[wave packet]] from a narrow range of frequencies will travel is called the [[group velocity]] and is determined from the [[gradient]] of the [[dispersion relation]]:
<math display="block">v_{\rm g} = \frac{\partial \omega}{\partial k}</math>

In almost all cases, a wave is mainly a movement of energy through a medium. Most often, the group velocity is the velocity at which the energy moves through this medium.

[[File:Light dispersion of a mercury-vapor lamp with a flint glass prism IPNr°0125.jpg|thumb|right|upright|Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a prism]]

Waves exhibit common behaviors under a number of standard situations, for example:

=== Transmission and media ===
{{Main|Rectilinear propagation|Transmittance|Transmission medium}}

Waves normally move in a straight line (that is, rectilinearly) through a ''[[transmission medium]]''. Such media can be classified into one or more of the following categories:
* A ''bounded medium'' if it is finite in extent, otherwise an ''unbounded medium''
* A ''linear medium'' if the amplitudes of different waves at any particular point in the medium can be added
* A ''uniform medium'' or ''homogeneous medium'' if its physical properties are unchanged at different locations in space
* An ''anisotropic medium'' if one or more of its physical properties differ in one or more directions
* An ''isotropic medium'' if its physical properties are the ''same'' in all directions

=== Absorption ===
{{Main|Absorption (acoustics)|Absorption (electromagnetic radiation)}}
Waves are usually defined in media which allow most or all of a wave's energy to propagate without [[Insertion loss|loss]]. However materials may be characterized as "lossy" if they remove energy from a wave, usually converting it into heat. This is termed "absorption." A material which absorbs a wave's energy, either in transmission or reflection, is characterized by a [[refractive index]] which is [[Complex number|complex]]. The amount of absorption will generally depend on the frequency (wavelength) of the wave, which, for instance, explains why objects may appear colored.

=== Reflection ===
{{Main|Reflection (physics)}}

When a wave strikes a reflective surface, it changes direction, such that the angle made by the [[incident ray|incident wave]] and line [[perpendicular|normal]] to the surface equals the angle made by the reflected wave and the same normal line.

=== Refraction ===
{{Main|Refraction}}

[[File:Wave refraction.gif|thumb|right|200 px|Sinusoidal traveling plane wave entering a region of lower wave velocity at an angle, illustrating the decrease in wavelength and change of direction (refraction) that results]]

Refraction is the phenomenon of a wave changing its speed. Mathematically, this means that the size of the [[phase velocity]] changes. Typically, refraction occurs when a wave passes from one [[Transmission medium|medium]] into another. The amount by which a wave is refracted by a material is given by the [[refractive index]] of the material. The directions of incidence and refraction are related to the refractive indices of the two materials by [[Snell's law]].

=== Diffraction ===
{{Main|Diffraction}}

A wave exhibits diffraction when it encounters an obstacle that bends the wave or when it spreads after emerging from an opening. Diffraction effects are more pronounced when the size of the obstacle or opening is comparable to the wavelength of the wave.

=== Interference ===
{{Main|Wave interference}}
[[Image:Two sources interference.gif|right|frame|Identical waves from two sources undergoing [[Interference (wave propagation)|interference]]. Observed at the bottom one sees 5 positions where the waves add in phase, but in between which they are out of phase and cancel.]]

When waves in a linear medium (the usual case) cross each other in a region of space, they do not actually interact with each other, but continue on as if the other one were not present. However at any point ''in'' that region the ''field quantities'' describing those waves add according to the [[superposition principle]]. If the waves are of the same frequency in a fixed [[phase (waves)|phase]] relationship, then there will generally be positions at which the two waves are ''in phase'' and their amplitudes ''add'', and other positions where they are ''out of phase'' and their amplitudes (partially or fully) ''cancel''. This is called an [[Interference (wave propagation)|interference pattern]].

=== Polarization ===
{{Main|Polarization (waves)}}

[[File:Circular.Polarization.Circularly.Polarized.Light Circular.Polarizer Creating.Left.Handed.Helix.View.svg|class=skin-invert-image|thumb|left]]

The phenomenon of polarization arises when wave motion can occur simultaneously in two [[orthogonal]] directions. [[Transverse wave]]s can be polarized, for instance. When polarization is used as a descriptor without qualification, it usually refers to the special, simple case of [[linear polarization]]. A transverse wave is linearly polarized if it oscillates in only one direction or plane. In the case of linear polarization, it is often useful to add the relative orientation of that plane, perpendicular to the direction of travel, in which the oscillation occurs, such as "horizontal" for instance, if the plane of polarization is parallel to the ground. [[Electromagnetic waves]] propagating in free space, for instance, are transverse; they can be polarized by the use of a [[polarizer|polarizing filter]].

Longitudinal waves, such as sound waves, do not exhibit polarization. For these waves there is only one direction of oscillation, that is, along the direction of travel.

=== Dispersion ===
[[File:Light dispersion conceptual waves.gif|thumb|right|270 px|Schematic of light being dispersed by a prism. Click to see animation.]]
{{Main|Dispersion relation|Dispersion (optics)|Dispersion (water waves)}}

Dispersion is the frequency dependence of the [[refractive index]],  a consequence of the atomic nature of materials.<ref name=hecht>{{Cite book |last=Hecht |first=Eugene |title=Optics |date=1998 |publisher=Addison-Wesley |isbn=978-0-201-83887-9 |edition=3 |location=Reading, Mass. Harlow}}</ref>{{rp|67}}
A wave undergoes dispersion when either the [[phase velocity]] or the [[group velocity]] depends on the wave frequency. Dispersion is seen by letting white light pass through a [[prism (optics)|prism]], the result of which is to produce the spectrum of colors of the rainbow. [[Isaac Newton]] was the first to recognize that this meant that white light was a mixture of light of different colors.<ref name=hecht/>{{rp|190}}

=== Doppler effect ===
{{main|Doppler effect}}
The Doppler effect or Doppler shift is the change in [[frequency]] of a wave in relation to an observer who is moving relative to the wave source.<ref name="Giordano">{{cite book
 | last1  = Giordano
 | first1 = Nicholas
 | title  = College Physics: Reasoning and Relationships
 | publisher = Cengage Learning
 | date   = 2009
 | pages  = 421–424
 | url    = https://books.google.com/books?id=BwistUlpZ7cC&pg=PA424
 | isbn   = 978-0534424718
 }}</ref> It is named after the [[Austria]]n physicist [[Christian Doppler]], who described the phenomenon in 1842.

== Mechanical waves ==
{{Main|Mechanical wave}}
A mechanical wave is an oscillation of [[matter]], and therefore transfers energy through a [[transmission medium|medium]].<ref>Giancoli, D. C. (2009) Physics for scientists & engineers with modern physics (4th ed.). Upper Saddle River, N.J.: Pearson Prentice Hall.</ref> While waves can move over long distances, the movement of the medium of transmission—the material—is limited. Therefore, the oscillating material does not move far from its initial position. Mechanical waves can be produced only in media which possess [[Elasticity (physics)|elasticity]] and [[inertia]]. There are three types of mechanical waves: [[transverse wave]]s, [[longitudinal wave]]s, and [[surface wave]]s.

=== Waves on strings ===
{{Main|String vibration}}
The transverse vibration of a string is a function of tension and inertia, and is constrained by the length of the string as the ends are fixed. This constraint limits the steady state modes that are possible, and thereby the frequencies.
The speed of a transverse wave traveling along a [[vibrating string]] (''v'') is directly proportional to the square root of the [[Tension (mechanics)|tension]] of the string (''T'') over the [[linear mass density]] (''μ''):
: <math> v = \sqrt{\frac{T}{\mu}}, </math>
where the linear density ''μ'' is the mass per unit length of the string.

=== Acoustic waves ===
{{Main|Acoustic wave}}

Acoustic or [[sound]] waves are compression waves which travel as body waves at the speed given by:
: <math> v = \sqrt{\frac{B}{\rho_0}}, </math>
or the square root of the adiabatic bulk modulus divided by the ambient density of the medium (see [[speed of sound]]).

=== Water waves ===
[[File:Shallow water wave.gif|thumb|right|400px]]

{{Main|Water waves}}
* [[ripple tank|Ripples]] on the surface of a pond are actually a combination of transverse and longitudinal waves; therefore, the points on the surface follow orbital paths.
* [[Sound]], a mechanical wave that propagates through gases, liquids, solids and plasmas.
* [[Inertial waves]], which occur in rotating fluids and are restored by the [[Coriolis effect]].
* [[Ocean surface wave]]s, which are perturbations that propagate through water.

=== Body waves ===
{{main|Body wave (seismology)}}
Body waves travel through the interior of the medium along paths controlled by the material properties in terms of density and modulus (stiffness). The density and modulus, in turn, vary according to temperature, composition, and material phase. This effect resembles the refraction of light waves. Two types of particle motion result in two types of body waves: Primary and Secondary waves.

=== Seismic waves ===
{{Main|Seismic wave}}
Seismic waves are waves of energy that travel through the Earth's layers, and are a result of earthquakes, volcanic eruptions, magma movement, large landslides and large man-made explosions that give out low-frequency acoustic energy. They include body waves—the primary ([[P wave]]s) and secondary waves ([[S wave]]s)—and surface waves, such as [[Rayleigh waves]], [[Love waves]], and [[Stoneley wave]]s.

=== Shock waves ===
[[File:Transonico-en.svg|class=skin-invert-image|thumb|right|300 px|Formation of a shock wave by a plane]]

{{Main|Shock wave}}

A shock wave is a type of propagating disturbance. When a wave moves faster than the local [[speed of sound]] in a [[fluid]], it is a shock wave. Like an ordinary wave, a shock wave carries energy and can propagate through a medium; however, it is characterized by an abrupt, nearly discontinuous change in [[pressure]], [[temperature]] and [[density]] of the medium.<ref>
{{citation
 | last = Anderson | first = John D. Jr.
 | title = Fundamentals of Aerodynamics | orig-year = 1984 | edition = 3rd
 | publisher =  [[McGraw-Hill|McGraw-Hill Science/Engineering/Math]] |date=January 2001
 | isbn = 978-0-07-237335-6
}}</ref>

{{See also|Sonic boom|Cherenkov radiation}}

=== Shear waves ===
{{main|Shear wave}}
Shear waves are body waves due to shear rigidity and inertia. They can only be transmitted through solids and to a lesser extent through liquids with a sufficiently high viscosity.

=== Other ===
* Waves of [[Traffic wave|traffic]], that is, propagation of different densities of motor vehicles, and so forth, which can be modeled as kinematic waves<ref name=Lighthill>{{cite journal |author1 = M.J. Lighthill | author1-link=James Lighthill |author2 = G.B. Whitham | author2-link=Gerald B. Whitham |year = 1955 |title = On kinematic waves. II. A theory of traffic flow on long crowded roads |journal = Proceedings of the Royal Society of London. Series A |volume = 229 | issue=1178 |pages = 281–345 |bibcode = 1955RSPSA.229..281L |doi = 10.1098/rspa.1955.0088 | citeseerx=10.1.1.205.4573 | s2cid=18301080 }}</ref><ref>{{cite journal |doi = 10.1287/opre.4.1.42 |author = P.I. Richards |year = 1956 |title = Shockwaves on the highway |journal = Operations Research |volume = 4 |issue = 1 |pages = 42–51 }}</ref>
* [[metachronal rhythm|Metachronal wave]] refers to the appearance of a traveling wave produced by coordinated sequential actions.

== Electromagnetic waves ==
[[File:Onde electromagnétique.png|class=skin-invert-image|thumb|right|300 px]]

{{Main|Electromagnetic wave}}
{{Further|Electromagnetic spectrum}}

An electromagnetic wave consists of two waves that are oscillations of the [[electric field|electric]] and [[magnetic field|magnetic]] fields. An electromagnetic wave travels in a direction that is at right angles to the oscillation direction of both fields. In the 19th century, [[James Clerk Maxwell]] showed that, in [[vacuum]], the electric and magnetic fields satisfy the [[wave equation]] both with speed equal to that of the [[speed of light]]. From this emerged the idea that [[visible light|light]] is an electromagnetic wave. The unification of light and electromagnetic waves was experimentally confirmed by [[Heinrich Hertz|Hertz]] in the end of the 1880s. Electromagnetic waves can have different frequencies (and thus wavelengths), and are classified accordingly in wavebands, such as [[radio waves]], [[microwaves]], [[infrared]], [[visible light]], [[ultraviolet]], [[X-rays]], and [[gamma ray]]s. The range of frequencies in each of these bands is continuous, and the limits of each band are mostly arbitrary, with the exception of visible light, which must be visible to the normal human eye.

== Quantum mechanical waves ==
{{Main|Schrödinger equation}}
{{See also|Wave function}}

=== Schrödinger equation ===
The [[Schrödinger equation]] describes the wave-like behavior of [[particle]]s in [[quantum mechanics]]. Solutions of this equation are [[wave function]]s which can be used to describe the probability density of a particle.

=== Dirac equation ===
The [[Dirac equation]] is a relativistic wave equation detailing electromagnetic interactions. Dirac waves accounted for the fine details of the hydrogen spectrum in a completely rigorous way. The wave equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-{{frac|1|2}} particles.

[[File:Wave packet (dispersion).gif|class=skin-invert-image|thumb|A propagating wave packet; in general, the ''envelope'' of the wave packet moves at a different speed than the constituent waves.<ref name=Fromhold>{{cite book |title = Quantum Mechanics for Applied Physics and Engineering |author = A.T. Fromhold |chapter = Wave packet solutions |pages = 59 ''ff'' |quote = (p. 61) ...the individual waves move more slowly than the packet and therefore pass back through the packet as it advances |chapter-url = https://books.google.com/books?id=3SOwc6npkIwC&pg=PA59 |isbn = 978-0-486-66741-6 |publisher = Courier Dover Publications |year = 1991 |edition = Reprint of Academic Press 1981 }}</ref>]]

=== de Broglie waves ===
{{Main|Wave packet|Matter wave}}

[[Louis de Broglie]] postulated that all particles with [[momentum]] have a wavelength
: <math>\lambda = \frac{h}{p},</math>
where ''h'' is the [[Planck constant]], and ''p'' is the magnitude of the [[momentum]] of the particle. This hypothesis was at the basis of [[quantum mechanics]]. Nowadays, this wavelength is called the [[de Broglie wavelength]]. For example, the [[electron]]s in a [[cathode-ray tube|CRT]] display have a de Broglie wavelength of about 10<sup>−13</sup>&nbsp;m.

A wave representing such a particle traveling in the ''k''-direction is expressed by the wave function as follows:
: <math>\psi (\mathbf{r}, \, t=0) = A e^{i\mathbf{k \cdot r}} , </math>
where the wavelength is determined by the [[wave vector]] '''k''' as:
: <math> \lambda = \frac {2 \pi}{k} , </math>
and the momentum by:
: <math> \mathbf{p} = \hbar \mathbf{k} . </math>

However, a wave like this with definite wavelength is not localized in space, and so cannot represent a particle localized in space. To localize a particle, de Broglie proposed a superposition of different wavelengths ranging around a central value in a [[wave packet]],<ref name=Marton>
{{cite book |title = Advances in Electronics and Electron Physics |page = 271 |chapter-url = https://books.google.com/books?id=g5q6tZRwUu4C&pg=PA271 |isbn = 978-0-12-014653-6 |year = 1980 |publisher = Academic Press |volume = 53 |editor1=L. Marton |editor2=Claire Marton |author = Ming Chiang Li |chapter = Electron Interference }}
</ref> a waveform often used in [[quantum mechanics]] to describe the [[wave function]] of a particle. In a wave packet, the wavelength of the particle is not precise, and the local wavelength deviates on either side of the main wavelength value.

In representing the wave function of a localized particle, the [[wave packet]] is often taken to have a [[Gaussian function|Gaussian shape]] and is called a ''Gaussian wave packet''.<ref name=wavepacket>{{cite book |url = https://books.google.com/books?id=7qCMUfwoQcAC&pg=PA60 |title = Quantum Mechanics |author1=Walter Greiner |author2=D. Allan Bromley |page = 60 |isbn = 978-3-540-67458-0 |edition = 2 |year = 2007 |publisher = Springer }}</ref><ref>{{cite book |title = Electronic basis of the strength of materials |author = John Joseph Gilman |url = https://books.google.com/books?id=YWd7zHU0U7UC&pg=PA57 |page = 57 |year = 2003 |isbn = 978-0-521-62005-5 |publisher = Cambridge University Press }}</ref><ref>{{cite book |title = Principles of quantum mechanics |author = Donald D. Fitts |url = https://books.google.com/books?id=8t4DiXKIvRgC&pg=PA17 |page = 17 |isbn = 978-0-521-65841-6 |publisher = Cambridge University Press |year = 1999 }}</ref> Gaussian wave packets also are used to analyze water waves.<ref name=Mei>{{cite book |url = https://books.google.com/books?id=WHMNEL-9lqkC&pg=PA47 |page = 47 |author = Chiang C. Mei |author-link=Chiang C. Mei |title = The applied dynamics of ocean surface waves |isbn = 978-9971-5-0789-3 |year = 1989 |edition = 2nd |publisher = World Scientific }}</ref>

For example, a Gaussian wavefunction ''ψ'' might take the form:<ref name="Bromley">
{{cite book |last1=Greiner |first1=Walter |url=https://books.google.com/books?id=7qCMUfwoQcAC&pg=PA60 |title=Quantum Mechanics |last2=Bromley |first2=D. Allan |publisher=Springer |year=2007 |isbn=978-3-540-67458-0 |edition=2nd |page=60}}
</ref>
: <math> \psi(x,\, t=0) = A \exp \left( -\frac{x^2}{2\sigma^2} + i k_0 x \right) , </math>
at some initial time ''t'' = 0, where the central wavelength is related to the central wave vector ''k''<sub>0</sub> as ''λ''<sub>0</sub> = 2π / ''k''<sub>0</sub>. It is well known from the theory of [[Fourier analysis]],<ref name=Brandt>
{{cite book |page = 23 |url = https://books.google.com/books?id=VM4GFlzHg34C&pg=PA23 |title = The picture book of quantum mechanics |author1=Siegmund Brandt |author2=Hans Dieter Dahmen |isbn = 978-0-387-95141-6 |year = 2001 |edition = 3rd |publisher = Springer }}
</ref> or from the [[Heisenberg uncertainty principle]] (in the case of quantum mechanics) that a narrow range of wavelengths is necessary to produce a localized wave packet, and the more localized the envelope, the larger the spread in required wavelengths. The [[Fourier transform]] of a Gaussian is itself a Gaussian.<ref name=Gaussian>
{{cite book |title = Modern mathematical methods for physicists and engineers |author = Cyrus D. Cantrell |page = [https://archive.org/details/modernmathematic0000cant/page/677 677] |url = https://archive.org/details/modernmathematic0000cant |url-access = registration |isbn = 978-0-521-59827-9 |publisher = Cambridge University Press |year = 2000 }}
</ref> Given the Gaussian:
: <math>f(x) = e^{-x^2 / \left(2\sigma^2\right)} , </math>
the Fourier transform is:
: <math>\tilde{ f} (k) = \sigma e^{-\sigma^2 k^2 / 2} . </math>

The Gaussian in space therefore is made up of waves:
: <math>f(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \ \tilde{f} (k) e^{ikx} \ dk ; </math>
that is, a number of waves of wavelengths ''λ'' such that ''kλ'' = 2 π.

The parameter σ decides the spatial spread of the Gaussian along the ''x''-axis, while the Fourier transform shows a spread in [[wave vector]] ''k'' determined by 1/''σ''. That is, the smaller the extent in space, the larger the extent in ''k'', and hence in ''λ'' = 2π/''k''.

[[File:GravitationalWave CrossPolarization.gif|class=skin-invert-image|thumb|right|Animation showing the effect of a cross-polarized gravitational wave on a ring of [[test particles]]]]

== Gravity waves ==
{{main|Gravity wave}}
Gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy works to restore equilibrium. Surface waves on water are the most familiar example.

== Gravitational waves ==

{{Main|Gravitational wave}}

[[gravitational radiation|Gravitational waves]] also travel through space. The first observation of gravitational waves was announced on 11 February 2016.<ref>{{cite web|title=Gravitational waves detected for 1st time, 'opens a brand new window on the universe'|url=http://www.cbc.ca/news/technology/ligo-gravitational-wave-1.3443697|publisher=Canadian Broadcasting Corporation|date=11 February 2016}}</ref>
Gravitational waves are disturbances in the curvature of [[spacetime]], predicted by Einstein's theory of [[general relativity]].

== See also ==
* [[Index of wave articles]]

=== Waves in general ===
{{columns-list|
* [[Mechanical wave]], in media transmission
* [[Wave equation]], general
* [[Wave interference]], a phenomenon in which two waves superpose to form a resultant wave
* [[Wave Motion (journal)|''Wave Motion'' (journal)]], a scientific journal
* [[Wavefront]], an advancing surface of wave propagation
}}

==== Parameters ====
{{columns-list|
* [[Frequency]]
* [[Phase (waves)]], offset or angle of a sinusoidal wave function at its origin
* [[Standing wave ratio]], in telecommunications
* [[Wavelength]]
* [[Wavenumber]]
}}

==== Waveforms ====
{{columns-list|
* [[Creeping wave]], a wave diffracted around a sphere
* [[Evanescent field]]
* [[Longitudinal wave]]
* [[Periodic travelling wave]]
* [[Sine wave]]
* [[Square wave]]
* [[Standing wave]]
* [[Transverse wave]]
}}

=== Electromagnetic waves ===
{{columns-list|
* [[Dyakonov surface wave]]
* [[Dyakonov–Voigt wave]]
* [[Earth–ionosphere waveguide]], in radio transmission
* [[Electromagnetic radiation]]
* [[Electromagnetic wave equation]], describes electromagnetic wave propagation
* [[Microwave]], a form of electromagnetic radiation
}}

=== In fluids ===
{{cols}}
* [[Airy wave theory]], in fluid dynamics
* [[Capillary wave]], in fluid dynamics
* [[Cnoidal wave]], in fluid dynamics
* [[Edge wave]], a surface gravity wave fixed by refraction against a rigid boundary
* [[Faraday wave]], a type of wave in liquids
* [[Gravity wave]], in fluid dynamics
* [[Internal wave]], a wave within a fluid medium
* [[Shock wave]], in aerodynamics
* Sound wave, a wave of [[sound]] through a medium such as air or water
* Tidal wave, a scientifically incorrect name for a [[tsunami]]
* [[Tollmien–Schlichting wave]], in fluid dynamics
* [[Wind wave]]
{{colend}}

=== In quantum mechanics ===
{{columns-list|
* [[Bloch's theorem]]
* [[Matter wave]]
* [[Pilot wave theory]], in Bohmian mechanics
* [[Wave function]]
* [[Wave packet]]
* [[Wave–particle duality]]
}}

=== In relativity ===
{{columns-list|
* [[Gravitational wave]], in relativity theory
* [[Relativistic wave equations]], wave equations that consider special relativity
* [[pp-wave spacetime]], a set of exact solutions to Einstein's field equation
}}

=== Other specific types of waves ===
{{cols}}
* [[Alfvén wave]], in plasma physics
* [[Atmospheric wave]], a periodic disturbance in the fields of atmospheric variables
* [[Fir wave]], a forest configuration
* [[Lamb waves]], in solid materials
* [[Rayleigh wave]], surface acoustic waves that travel on solids
* [[Spin wave]], in magnetism
* [[Spin density wave]], in solid materials
* [[Trojan wave packet]], in particle science
* [[Waves in plasmas]], in plasma physics
{{colend}}

=== Related topics ===
{{cols}}
* [[Absorption (electromagnetic radiation)]]
* [[Antenna (radio)]]
* [[Beat (acoustics)]]
* [[Branched flow]]
* [[Cymatics]]
* [[Diffraction]]
* [[Dispersion (water waves)]]
* [[Doppler effect]]
* [[Envelope detector]]
* [[Fourier transform]] for computing periodicity in evenly spaced data
* [[Group velocity]]
* [[Harmonic]]
* [[Huygens–Fresnel principle]]
* [[Index of wave articles]]
* [[Inertial wave]]
* [[Least-squares spectral analysis]] for computing periodicity in unevenly spaced data
* [[List of waves named after people]]
* [[Phase velocity]]
* [[Photon]]
* [[Polarization (physics)]]
* [[Propagation constant]]
* [[Radio propagation]]
* [[Ray (optics)]]
* [[Reaction–diffusion system]]
* [[Reflection (physics)]]
* [[Refraction]]
* [[Resonance]]
* [[Ripple tank]]
* [[Rogue wave]]
* [[Scattering]]
* [[Shallow water equations]]
* [[John N. Shive#Shive wave machine|Shive wave machine]]
* [[Sound]]
* [[Standing wave]]
* [[Transmission medium]]
* [[Velocity factor]]
* [[Wave equation]]
* [[Wave power]]
* [[Wave turbulence]]
* [[Wind wave]]
* [[Wind wave#Formation]]
{{colend}}

== References ==
{{reflist}}

== Sources ==
{{refbegin}}
* {{cite book |last1 = Fleisch |first1 = D. |last2 = Kinnaman |first2 = L. |year = 2015 |title = A student's guide to waves |location = Cambridge|publisher = Cambridge University Press |isbn = 978-1107643260 |bibcode = 2015sgw..book.....F }}
* {{cite book|last1=Campbell|first1=Murray|title=The musician's guide to acoustics|year=2001|publisher=Oxford University Press|location=Oxford|isbn=978-0198165057|edition=Repr.|last2=Greated |first2=Clive}}
* {{cite book |first = A.P. |last = French |title = Vibrations and Waves (M.I.T. Introductory physics series) |publisher = Nelson Thornes |year = 1971 |isbn = 978-0-393-09936-2 |oclc = 163810889 }}
* {{cite book |last = Hall |first = D.E. |year = 1980 |title = Musical Acoustics: An Introduction |location = Belmont, CA |publisher = Wadsworth Publishing Company |isbn = 978-0-534-00758-4 }}.
* {{cite book|last=Hunt|first=Frederick Vinton |title=Origins in acoustics.|year=1978|publisher=Published for the Acoustical Society of America through the American Institute of Physics|location=Woodbury, NY|isbn=978-0300022209}}
* {{cite book |last1 = Ostrovsky |first1 = L.A. |last2 = Potapov |first2 = A.S. |year = 1999 |title = Modulated Waves, Theory and Applications |location = Baltimore |publisher = The Johns Hopkins University Press |isbn = 978-0-8018-5870-3 }}.
* {{Cite book | publisher = Academic Press | isbn = 9780123846532 | last1 = Griffiths | first1 = G. | first2 = W.E. | last2 = Schiesser | title = Traveling Wave Analysis of Partial Differential Equations: Numerical and Analytical Methods with Matlab and Maple | year = 2010 }}
* Crawford jr., Frank S. (1968).  ''Waves (Berkeley Physics Course, Vol. 3)'', McGraw-Hill,  {{ISBN|978-0070048607}} [https://archive.org/details/Waves_371 Free online version]
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{{refend}}

== External links ==
{{Sister project links| wikt=wave | commons=Wave | b=no | n=no | q=Wave | s=no | v=no | voy=no | species=no | d=no}}
* [https://feynmanlectures.caltech.edu/I_51.html The Feynman Lectures on Physics: Waves]
* [http://www.scholarpedia.org/article/Linear_and_nonlinear_waves Linear and nonlinear waves]
* [https://www.scienceaid.net/physics/waves/properties.html Science Aid: Wave properties – Concise guide aimed at teens] {{Webarchive|url=https://web.archive.org/web/20190904042222/https://www.scienceaid.net/physics/waves/properties.html |date=2019-09-04 }}
* [https://www.youtube.com/watch?v=DovunOxlY1k "AT&T Archives: Similiarities of Wave Behavior"] demonstrated by J.N. Shive of Bell Labs (video on [[YouTube]])

{{Velocities of Waves}}
{{Patterns in nature}}

{{Authority control}}

[[Category:Waves| ]]
[[Category:Differential equations]]
[[Category:Articles containing video clips]]